Optimal. Leaf size=27 \[ \frac {\left (-a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1482, 643}
\begin {gather*} \frac {\left (-a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 643
Rule 1482
Rubi steps
\begin {align*} \int x^2 \left (b+2 c x^3\right ) \left (-a+b x^3+c x^6\right )^p \, dx &=\frac {1}{3} \text {Subst}\left (\int (b+2 c x) \left (-a+b x+c x^2\right )^p \, dx,x,x^3\right )\\ &=\frac {\left (-a+b x^3+c x^6\right )^{1+p}}{3 (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 27, normalized size = 1.00 \begin {gather*} \frac {\left (-a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 26, normalized size = 0.96
method | result | size |
gosper | \(\frac {\left (c \,x^{6}+b \,x^{3}-a \right )^{1+p}}{3+3 p}\) | \(26\) |
risch | \(-\frac {\left (-c \,x^{6}-b \,x^{3}+a \right ) \left (c \,x^{6}+b \,x^{3}-a \right )^{p}}{3 \left (1+p \right )}\) | \(38\) |
norman | \(-\frac {a \,{\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}-a \right )}}{3 \left (1+p \right )}+\frac {b \,x^{3} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}-a \right )}}{3+3 p}+\frac {c \,x^{6} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}-a \right )}}{3+3 p}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 37, normalized size = 1.37 \begin {gather*} \frac {{\left (c x^{6} + b x^{3} - a\right )} {\left (c x^{6} + b x^{3} - a\right )}^{p}}{3 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 37, normalized size = 1.37 \begin {gather*} \frac {{\left (c x^{6} + b x^{3} - a\right )} {\left (c x^{6} + b x^{3} - a\right )}^{p}}{3 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.85, size = 25, normalized size = 0.93 \begin {gather*} \frac {{\left (c x^{6} + b x^{3} - a\right )}^{p + 1}}{3 \, {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.08, size = 52, normalized size = 1.93 \begin {gather*} {\left (c\,x^6+b\,x^3-a\right )}^p\,\left (\frac {b\,x^3}{3\,p+3}-\frac {a}{3\,p+3}+\frac {c\,x^6}{3\,p+3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________